Some examples relating the deleted product to imbeddability

Ummel, Brian R.

doi:10.1090/s0002-9939-1972-0290349-0

Cited by 6 publications

(4 citation statements)

References 5 publications

(4 reference statements)

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“…Abrams gave a similar result for two points in K 5 and the complete bipartite graph K 3,3 ; these spaces are homotopy equivalent to Σ 6 and Σ 4 respectively [Abr00]. That these are the only two-strand braid groups to give a surface group was shown by Ummel [Umm72].…”

Section: Braid Groupsmentioning

confidence: 84%

Models for configuration space in a simplicial complex

Wiltshire-Gordon

2019

Colloq. Math.

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We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve y 2 z = x 3 + x 2 z, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its twoand three-strand local braid groups near the singular point.

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Section: Braid Groupsmentioning

confidence: 84%

Models for configuration space in a simplicial complex

Wiltshire-Gordon

2019

Colloq. Math.

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“…Arguments which might relate to a possible common higher dimensional generalization of the Kuratowski and Robertson-Seymour-Thomas theorems could include, apart from those in §6, those in [68], [69] (see also [81]), [73]. Unfortunately, the present paper gives little clue to understanding proofs of the Kuratowski and Robertson-Seymour-Thomas theorems, but it does attempt to provide a better understanding of their statements.…”

Section: A Graphsmentioning

confidence: 94%

Combinatorics of embeddings

Melikhov¹

2011

Preprint

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We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex dichotomial if to every nonempty cell there corresponds a unique nonempty cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K 5 and K 3,3 ; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family.In higher dimensions n ≥ 3, we observe that in order to characterize those compact n-polyhedra that embed in R 2n in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n − 1)-connected n-dimensional finite cell complexes that do not embed in R 2n yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n + 1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i + 2)-simplices) at least ten non-simplicial complexes.

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“…We should mention that it is well known that the early literature on the deleted product cohomology contained a number of errors. Errors in [20] were observed by Ummel [29] and Barnett and Farber [6]. An error in [10] was discussed by Sarkaria [23] and Barnett and Farber [6].…”

Section: Deleted Product Cohomology and The Van Kampen-wu Invariantmentioning

confidence: 99%

Bad drawings of small complete graphs

Cairns¹,

Groves²,

Nikolayevsky³

2019

Preprint

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We show that for K 5 (resp. K 3,3 ) there is a drawing with i independent crossings, and no pair of independent edges cross more than once, provided i is odd with 1 ≤ i ≤ 15 (resp. 1 ≤ i ≤ 17). Conversely, using the deleted product cohomology, we show that for K 5 and K 3,3 , if A is any set of pairs of independent edges, and A has odd cardinality, then there is a drawing in the plane for which each element in A cross an odd number of times, while each pair of independent edges not in A cross an even number of times. For K 6 we show that there is a drawing with i independent crossings, and no pair of independent edges cross more than once, if and only if 3 ≤ i ≤ 40.

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Some examples relating the deleted product to imbeddability

Cited by 6 publications

References 5 publications

Models for configuration space in a simplicial complex

Models for configuration space in a simplicial complex

Combinatorics of embeddings

Bad drawings of small complete graphs

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